One-one function

Step 1: In a one-one function, two different inputs cannot have the same output. Step 2: So different inputs must give different outputs. Step 3: To test one-one, assume (f(a)=f(b)) and show (a=b).

Step 1: Here (2) and (3) are different inputs. Step 2: Both give the output (5), so the function is not one-one. Step 3: In a one-one function, equal outputs must come from equal inputs.

Step 1: Assume (f(a)=f(b)). Step 2: (2a+1=2b+1) gives (a=b). Step 3: A linear function (ax+b) is one-one when (a\neq0).

Step 1: This is a constant function because its value is (7) for every (x). Step 2: (f(1)=7) and (f(2)=7), although (1\neq2). Step 3: A constant function on a larger domain is not one-one.

Step 1: In a modulus function, (x) and (-x) can give the same value. Step 2: (f(2)=2) and (f(-2)=2), although (2\neq-2). Step 3: For modulus functions, check opposite-sign inputs.

Step 1: (e^x) is an increasing function. Step 2: As (x) increases, (e^x) also increases, so two different inputs cannot give the same output. Step 3: Increasing functions are usually easy to test for one-one.

Step 1: (\log x) is increasing on its domain ((0,\infty)). Step 2: Different positive inputs give different logarithmic values. Step 3: For logarithmic functions, remember that the domain contains positive numbers only.

Step 1: From (f(a)=f(b)), we get (3a-5=3b-5). Step 2: Simplifying gives (3a=3b), hence (a=b). Step 3: The condition (f(a)=f(b)\Rightarrow a=b) is the main test for one-one.

Step 1: To show a function is not one-one, show equal outputs for different inputs. Step 2: (f(1)=2) and (f(-1)=2), while (1\neq-1). Step 3: Finding only one value is not enough; compare two different inputs.

Step 1: Adding (4) to different natural numbers gives different numbers. Step 2: If (n_1+4=n_2+4), then (n_1=n_2). Step 3: Adding the same number does not destroy one-one property.

Step 1: Doubling different integers gives different results. Step 2: From (2n_1=2n_2), we get (n_1=n_2). Step 3: A linear rule with non-zero coefficient remains one-one.

Step 1: The integers include both (1) and (-1). Step 2: (f(1)=1) and (f(-1)=1), so different inputs have the same output. Step 3: The square function is not one-one on all integers.

Step 1: Each element of the domain has exactly one image. Step 2: The images (4,5,6) are all distinct. Step 3: For finite sets, one-one can be checked quickly by listing the images.

Step 1: (1) and (2) are different elements. Step 2: Both have image (4), so the function is not one-one. Step 3: For finite sets, repeated images mean the function is not one-one.

Step 1: A one-one function needs a distinct image for each domain element. Step 2: Here (5) distinct images are needed, but the codomain has only (3) elements. Step 3: For a one-one function, the codomain must have at least as many elements as the domain.

Step 1: A one-one function needs (3) distinct images. Step 2: The codomain has (5) elements, so choosing (3) distinct images is possible. Step 3: For finite sets, compare the number of elements first.

Step 1: Assume (f(a)=f(b)). Step 2: (-4a+9=-4b+9) gives (a=b). Step 3: A negative coefficient does not destroy one-one property; the coefficient just must not be zero.

Step 1: The value of (0x+6) is (6) for every (x). Step 2: (f(1)=6) and (f(5)=6), although (1\neq5). Step 3: In a linear expression, if the coefficient of (x) is (0), the function becomes constant.

Step 1: (x^3) is an increasing function on real numbers. Step 2: If (a^3=b^3), then (a=b). Step 3: The basic odd-power cubic function is one-one.

Step 1: (1) and (-1) are different real numbers. Step 2: Both give the value (1) under (x^4). Step 3: Even-power functions are generally not one-one on the whole real domain.

Step 1: (f(4)=14) and (f(5)=15). Step 2: Different inputs give different values. Step 3: In such a linear function, adding the same number preserves order and one-one property.

Step 1: In a one-one function, different inputs have different images. Step 2: Since (a_1\neq a_2), we must have (f(a_1)\neq f(a_2)). Step 3: This is the direct meaning of one-one.

Step 1: The condition for one-one connects equal outputs to equal inputs. Step 2: If (f(a_1)=f(a_2)), then (a_1=a_2). Step 3: This is the most commonly used form in proofs.

Step 1: For one-one, all images must be distinct. Step 2: Here all four images are given as distinct. Step 3: In finite functions, no repetition of images means the function is one-one.

Step 1: Such repetition is not allowed in a one-one function. Step 2: If two different inputs have the same image, the condition fails. Step 3: One counterexample is enough to disprove one-one.